Solving Systems of Equations Using Multiple Strategies NCTM Interactive Institute, 2015

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Common Core Standards

This session will address the following:

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8.EE.8a Understand that solutions to a system of two linear equationsin two variables correspond to points of intersection of theirgraphs, because points of intersection satisfy both equationssimultaneously.

8.EE.8b Solve systems of two linear equations in two variablesalgebraically, and estimate solutions by graphing the equations.Solve simple cases by inspection.

8.EE.8c Solve real-world and mathematical problems leading to two linearequations in two variables.

Warm Up

Find two different equations that are satisfied by the point (5, -2). Describe the relationship of the graphs of your equations.

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Warm Up Discussion

• What strategies did you use to find two equations that satisfy the point (5, -2)?

• What are some other strategies we could have used?

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Going Bananas!

Consider. . .

A banana peel weighs one-eighth the total weight of a banana. If an unpeeled banana balances a peeled banana of the same weight plus seven-eighths of an ounce, how much does a banana weigh with peel?

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Going Bananas

What did you have to understand in order to solve this problem?

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Consider this student diagram. . .

7Maida, P. (2004). Using algebra without realizing it.

Mathematics Teaching in the Middle School. 9(9).

Chicken Problem

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Chicken Problem Solutions

What did you notice about students’ solutions?

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Solving Systems of Equation

What strategies can be used to solve systems of equations?

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Strategies

• Elimination

• Substitution

• Graphing

• Calculator

• Manipulatives

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Consider

When is one strategy “better” than another?

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Sequencing Instruction

Think about the instructional sequence you use in teaching how to solve systems of equations.

What do students do first in the lessons?

How do the lessons change over time?

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Knotty Problem

In groups, you will use the meter stickand rope to complete the Knotty Problem.

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Knotty Problem

• What observations and patterns have you noticed?

• What does the y-intercept represent?

• What does the slope represent?

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Knotty Problem - extension

• Pair up with another group, and exchange “rules”.

• Make a prediction:

Assuming each group started with equal lengths of rope, how many knots could you make in your rope so that it’s as close as possible to the same length as the rope from your partnering group with three knots? Use, pictures, words, and symbols to describe your solution.

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Fish Problem

How much does a fish weigh if its tail weights 4 kilograms, its head weighs as much as its tail and half its body, and its body weighs as much as its head and tail together?

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Fish Problem

• What were your initial thoughts about a strategy to solve the Fish Problem?

• Was this strategy useful?

• What other strategies could you have used?

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Reflection

(Principles to Actions: Ensuring Mathematical Success for All [NCTM 2014], p. 47)

Reflection

(Principles to Actions: Ensuring Mathematical Success for All [NCTM 2014], p. 48)

How were you engaged in the Mathematical Practices?

1. Make sense of problems and persevere in solving them.

2. Reason abstractly and quantitatively.

3. Construct viable arguments and critique the reasoning of others.

4. Model with mathematics. 5. Use appropriate tools

strategically. 6. Attend to precision. 7. Look for and make use of

structure. 8. Look for and express

regularity in repeated reasoning.

• Banana Problem

• Chicken Problem

• Knotty Problem

• Fish Problem

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Exit Ticket: Reflection

• What new idea(s) do you want to implement into your classroom?

• What challenges did you encounter during this session?

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Disclaimer

The National Council of Teachers of Mathematics is a public voice of mathematics education, providing vision, leadership, and professional development to support teachers in ensuring equitable mathematics learning of the highest quality for all students. NCTM’s Institutes, an official professional development offering of the National Council of Teachers of Mathematics, supports the improvement of pre-K-6 mathematics education by serving as a resource for teachers so as to provide more and better mathematics for all students. It is a forum for the exchange of mathematics ideas, activities, and pedagogical strategies, and for sharing and interpreting research. The Institutes presented by the Council present a variety of viewpoints. The views expressed or implied in the Institutes, unless otherwise noted, should not be interpreted as official positions of the Council.

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